I'm trying to help someone pass their final exam (analysis of functions) so they can graduate high school and move on to college. (Not a teacher, just another student, currently in high level math)
They are currently struggling with many concepts, most recently including
She is very frustrated because she feels like she needs to know what
ln is and how to find it by hand and how to do everything by hand. If she uses a calculator, she feels like she doesn't understand and isn't learning the math. (In our specific problem, I told her she needed a calculator to enter ln(5.4) and take that divided by 8 for her answer.) Her teacher is far less than helpful (I'm not convinced her teacher even understands it), leaving her more frustrated.
I explained to her that
ln is a function, like an oven where you bake a cake. You put something in, and depending what you put in, you get something else out. The calculator is just the oven, and
ln is just a recipe. You are still the one doing the baking.
She asked what people did before calculators and I told her that before calculators, you used a slide rule, and before that, you used a log table. Nobody ever did them by hand. I compared it to deriving pi by hand; possible, but pointless.
She wanted to know where
ln comes from and what it meant, so I met her with this wonderful ms paint diagram:
This seemed to help her understand finally where
ln comes from, after I also explained that
ln is just log base e.
But she was still frustrated that she couldn't do it by hand and didn't know how to find the area. I warned her that this is literally second semester college calculus but she seemed to need to know, so I briefly explained to her the infinite series used to calculate it as simply as possible, which I think she actually understood. She asked why she couldn't do it by hand, and I told her the obnoxious number of iterations she would have to do.
I think I finally got the concept of
ln through to her, but I fear she will have similar problems with the rest of her work with logs, and upcoming sigma notation (for many iterations, not wanting to do it in the n-spire), and other things. How can I convince her that the mechanics of the computations aren't what are important here?
I tried explaining that these questions will be answered in higher math, but seems unable (unwilling?) to grasp that understanding the chemistry of baking isn't needed to know how to make a good cake.
It is worse because, asking why she needs to know this, I really don't have a good answer. Most of math is practical, but I was always confused as to why we teach lns and e without the proper calculus background to really understand them.